Superfícies de Weingarten regradas em geometrias de Thurston

Abstract

In this work, we classify all the ruled Weingarten surfaces of class C³ immersed in the Heisenberg group Nil_3, obtaining explicit parametrizations for each case. We also classify translation surfaces of the graph type in Nil_3 which are linear-Weingarten or whose norm of the second fundamental form is constant. We also classify ruled Weingarten surfaces in the product spaces, S²×R and H²×R. In the space SL_2(R)~, we did an initial study in which we classified vertical and horizontal ruled surfaces whose one of the curvatures, either mean or gaussian, is constant. We included yet a chapter considering Bishop surfaces (or Monge surfaces) in R³, in which we classified all such surfaces of class C² satisfying some polynomial Weingarten relation.